Partial correlation networks of Gaussian processes

TL;DR

This paper introduces a novel framework for understanding direct relationships in multivariate Gaussian processes using process-level partial correlation, extending graphical model concepts to spatial data.

Contribution

It establishes a characterization of process-level partial correlation for Gaussian processes, including nonstationary models, clarifying limitations of existing spatial modeling approaches.

Findings

Partial cross-correlation functions factorize into a coefficient and an attenuation term.

Spectrally inside-out processes include the multivariate Matérn and coregionalization models.

The framework provides necessary and sufficient conditions for conditional independence in spatial processes.

Abstract

In Gaussian graphical models, conditional independence and partial correlations are natural inferential targets for understanding direct relationships in multivariate data. No comparable framework exists for spatial processes, where multivariate analysis defaults to modeling unconditional cross-covariance structure, even when direct relationships remain of scientific interest. We address this gap by establishing a novel characterization of process-level partial correlation for multivariate Gaussian processes that recovers a direct link with Gaussian graphical models. Our analysis proceeds through a class of stationary multivariate processes, termed spectrally inside-out, in which a precision matrix modulates the strength of conditional dependence and yields necessary and sufficient conditions for conditional independence. Within this class, partial cross-correlation functions factorize into a process-level partial correlation coefficient and an attenuation term independent of cross-process parameters. The spectrally inside-out class includes the separable coregionalization model, a process convolution construction, and the parsimonious multivariate Mat\'ern, for which such a characterization was previously thought unavailable. We further show that a nonstationary inside-out model satisfies the same factorization and admits the same necessary and sufficient conditions. Our results clarify the limitations of existing approaches: linear coregionalization models encode conditional independence through the zero pattern of the inverse factor loading matrix and do not result in interpretable partial cross-correlation functions. Low-rank spatial factor models lack a meaningful graphical characterization. Methods that enforce network structure through auxiliary graphical layers only characterize presence or absence of graph edges. We illustrate our results through synthetic and real data.